Resonant Mode Shapes Research Articles

High-efficient internal resonance energy harvesting: Modelling and experimental study

This research investigates a U-shaped vibration-based energy harvester with broadband and bi-directional features via an internal resonance. The proposed U-shaped structure has been tailored in a way that displays a two-to-one internal resonance between the selected transverse modes. The resonant modes and mode shapes of the proposed device have been designed and validated through an analytical method, the finite-element simulations as well as experimental tests, showing good agreement. The experimental tests imply the benefits of energy transfer between two modes, as resonance phenomena in all segments have been successfully triggered to obtain large-amplitude oscillations for high-efficient energy conversion via piezoelectric effects. Under harmonic excitations, different from conventional multi-mode vibration energy harvesters, the proposed device presented broadened resonance regions for both modes, and exhibited equally effective bandwidths under forward/backward sweeps. Moreover, the device is capable of being excited in two orthogonal directions as a bi-directional energy harvester. Parametric studies on the acceleration levels are also presented to evaluate the system performance and reveal the critical excitation level for triggering the internal resonance. It has been found that even under a 0.1 g acceleration level, the internal resonance phenomenon can still be observed between the first and second modes, which implies the feasibility of the proposed device even for ambient vibration sources with low energies.

Achievable accuracy of resonating nanomechanical systems for mass sensing of larger analytes in GDa range

Measurement of larger analytes such as many chemical and biological structures or viruses in gigadalton (GDa) range is a reminding fundamental task in analytical chemistry and life sciences, which can be possibly resolved with the resonating nanomechanical systems. Common approaches to mass sensing with these systems model the bound analyte as a point particle and assume the analyte does not change the vibrational mode shapes. However, for larger analytes their stiffness and size not only affect the resonant frequencies but also cause the significant changes in the vibrational mode shapes making their measurement highly challenging and still under-explored problem. Here, we develop a 3D model capable to accurately predict the resonant frequencies and vibrational mode shapes of the resonating nanomechanical systems with the bound analyte of arbitrary properties and size. Then, we examine in details the impact of analyte properties, size and its position of attachment on the resonant frequencies and vibrational mode shapes and, correspondingly, resolve a dispute over the achievable detection limits of the nanomechanical systems, especially for mass sensing of larger analytes. Furthermore, we identify three different sensing regimes, that is, the ultra-light, light and heavy, for which the effects of the analyte mass, stiffness, its size and the position of attachment on the accuracy of the determined mass can be separated. For the ultra-light regime (mass ratio < 10−3) the analyte stiffness and its size affect notably the resonant frequencies, the point mass approximation is inaccurate and the analyte can be identified by its mass and stiffness. For the light regime the point mass approximation is accurate, while for the heavy one (mass ratio > 2∙10−2) the mass effect dominates the frequency response and alters the vibrational mode shapes which, for the common approaches, yields the errors in the determined analyte mass. Finally, we also propose an easily accessible approach for the identification of larger analytes (in GDa range) that does not require the advanced computational tools or experimental setup, applicable to the majority of the clamped-clamped ends resonating nanomechanical systems.

Vibration analysis of MEMS vibrating mesh atomizer

Vibrating mesh atomizers (VMAs) are increasing in demand for various applications that require high quality droplet size distribution of aerosols. However, manufacturing limitations of metallic mesh atomizers have prevented researchers from investigating the dynamics and vibration analysis required to further enhance performance. Newly developed MEMS based VMAs allow these devices to be custom designed including varying aperture size, shape, and pitch as well as varying membrane dimensions. In this paper, a systematic vibration analysis of silicon-based MEMS based VMA was investigated to better understand the mechanisms of the atomization process and atomization rate. The MEMS atomizer consists of a microfabricated mesh on silicon membrane coupled with piezoelectric ring. The atomization process with this device is intricate to model due to combination of fluid transfer and dynamics of the membrane actuated by the piezoelectric ring. This paper uses multiphysics finite element modeling validated by experimental analysis to better understand the dynamics of the membrane and key parameters that affect the vibration analysis and atomization process. Resonance frequency, displacement, velocity, and mode shapes of the various dynamic modes of the atomizer were studied using finite element analysis and compared with the experimental results to validate the model. The results demonstrate a strong correlation between the modeled and experimental results of the resonant frequencies and atomization rates. The results can be used to design VMAs with enhanced performance for specific applications in the future.

Design and development of a novel piezoelectrically actuated asymmetrical flexible microgripper

This paper presents the design, modeling, optimization, simulation, and experimental study of a new asymmetric flexible microgripper driven by a piezoelectric actuator. By comparing the asymmetric microgripper with the corresponding symmetric one, new guideline is introduced to design the gripper mechanism. The uniqueness of the gripper lies in consistent working mode and first resonant mode shapes without the dense mode of resonance. The proposed asymmetric gripper is composed of a three-stage flexure amplification mechanism, which can achieve friction-free and clearance-free displacement amplification. The gripper adopts the separate design of driving arm and sensing arm to avoid possible interference between them. The key geometric parameters affecting the performance of the flexure hinge are selected and optimized. By resorting to the pseudo-rigid body model, the kinematic, static, and dynamic models of the asymmetric gripper are derived. The working stroke, clamping force, and resonant frequency of the designed monolithic gripper are verified by conducting both simulations and experiments. A variety of experiments on the successful gripping of micro-objects of different sizes and shapes demonstrate its promising use in practical micromanipulation and microassembly applications.

Influence of residual stress and texture on the resonances of polycrystalline metals.

Efficient nondestructive qualification of additively manufactured (AM) metallic parts is vital for the current and future adoption of AM parts throughout several industries. Resonant ultrasound spectroscopy (RUS) is a promising method for the qualification and characterization of AM parts. Although the adoption of RUS in this setting is emerging, the influence of residual stress and texture, which are both very common in AM parts, is not well understood. In this article, a stress- and texture-dependent constitutive relation is used to study the influence on free vibrational behavior in a RUS setting. The results that follow from using the Rayleigh-Ritz method and finite element analysis suggest that residual stress and texture have a significant impact on the resonance frequencies and mode shapes. These results support the potential of using RUS to sense texture and residual stress in AM parts. Additionally, these results suggest that RUS measurements could be misinterpreted when the stress and texture are not accounted for, which could lead to a false positive/negative diagnosis when qualifying AM parts.

Design of a radial ultrasonic horn for plastic welding using finite element analysis

In this paper, a radial ultrasonic horn is designed by finite element analysis. Three different modes are obtained corresponding to specific natural frequencies. The distribution of amplitude and stresses of resonant mode shape are investigated. The 7075 aluminum grade is utilized for fabricating the radial ultrasonic horn. The ultrasonic sewing machine using the radial horn is established. The measured frequencies of the simulated and machined results are compared. Two kinds of polyvinyl chloride (PVC) tarpaulin fabrics with densities of 220 g sqm−1 and 320 g sqm−1 are welded by the radial ultrasonic welding system. Two patterns of the rollers are utilized for testing the welding ability of the machine. The ultrasonic power, air pressure, and the velocity of the roller are set at 1500 W, 3 kg cm−2, and 6 m min−1, respectively. Good welding joints are achieved. The tensile strengths of welding joints of two tarpaulin sheets are measured as 144.9 g mm−1 and 134.6 g mm−1, respectively.

A tuned mass amplifier for enhanced haptic feedback

Vibro-tactile feedback is, by far the most common haptic interface in wearable or touchable devices. This feedback can be amplified by controlling the wave propagation characteristics in devices, by utilizing phenomena such as structural resonance. However, much of the work in vibro-tactile haptics has focused on amplifying local displacements in a structure by increasing local compliance. In this paper, we show that engineering the resonance mode shape of a structure with embedded localized mass amplifies the displacements without compromising on the stiffness or resonance frequency. The resulting structure, i.e., a tuned mass amplifier, produces higher tactile forces (7.7 times) compared to its counterpart without a mass, while maintaining a low frequency. We optimize the proposed design using a combination of a neural network and sensitivity analysis, and validate the results with experiments on 3-D printed structures. We also study the performance of the device on contact with a soft material, to evaluate the interaction with skin. Potential avenues for future work are also presented, including small form factor wearable haptic devices and remote haptics.

Resonance Risk and Mode Shape Management in the Frequency Domain to Prevent Squeak and Rattle

Abstract Avoiding quality problems in passenger cars, such as squeak and rattle (S&amp;R), has been a remarkable cost-saving consideration. The introduction of electric engines and autonomous driving is expected to further stress the need for quieter cabins. However, the complexity of S&amp;R events has obstructed the practical treatment of these quality issues in the pre-design-freeze phases of product development. In this study, new quantified frequency-domain metrics are proposed to measure the risk of S&amp;R generation in car subsystems. The proposed metrics measure the resonance risk and the mode shape similarity in the critical interfaces for S&amp;R. The calculations are done based on the system response in the frequency domain. Compared with the time-domain evaluation methods, the knowledge about the system excitation levels is not essential and the calculations are more time-efficient. The proposed metrics can be used in design optimization processes to involve S&amp;R attributes in the pre-design-freeze attribute trade-off activities besides other attributes. In this work, these metrics were used in a previously developed two-stage optimization approach to determine the connection configuration in two industrial cases. As compared with the baseline design, the risk for S&amp;R was reduced by improving the system behavior in terms of resonance risk and mode shape similarity. This was achieved by applying adjustments to the location of the fasteners while maintaining the same general connection configuration concept.

A novel semi-active actuator with tunable mass moment of inertia for noise control applications

Semi-active control approaches have been developed and employed for a variety of vibration-reduction applications, including adaptive vehicle suspensions, earthquake protection for civil structures, and more. These semi-active systems require significantly less energy than the corresponding active solutions, while continuing to provide a high level of performance, therefore they are an attractive approach. In the context of noise reduction barriers, semi-active control often refers to piezoelectric patches or stacks connected to shunt electric circuits in order to transform and dissipate mechanical energy. However, a different type of semi-active actuators can also be used to adapt the mechanical features of noise barriers in order to deal with varying noise properties. This paper proposes a novel semi-active actuator with tunable mass moment of inertia. When attached to a noise barrier, it can alter the resonant frequencies and mode shapes of the barrier in order to reduce acoustic radiation at dominant frequencies in the noise. As the presented results show, this actuator can enhance the transmission loss of a noise barrier for time-varying narrow-band noise by more than 10 dB in targeted frequency bands. Alternatively, the proposed actuator can be used to optimize acoustic radiation from a panel acting as a sound source. Both scenarios are considered and analyzed in this paper by employing mathematical modeling, experimental validation and numerical investigation.

A Theoretical Model for Analyzing the Thickness-Shear Vibration of a Circular Quartz Crystal Plate With Multiple Concentric Ring Electrodes.

A dynamic model to analyze the thickness-shear vibration of a circular quartz crystal plate with multiple concentric ring electrodes on its upper and bottom surfaces is established with the aid of coordinate transformation. The theoretical solution is obtained, which can be written in a superposition form of Mathieu functions and modified Mathieu functions. The convergence of the solution is demonstrated, and the correctness is numerically validated via results from the finite element method (FEM). Subsequently, a systematic investigation is carried out to quantify the effect of the electrode size on the energy trapping phenomenon, i.e., the resonant frequency and mode shape, which reveals that the ring electrode has a great influence on the work performance of resonators. With the increase of the electrode inertia, i.e., the radius and mass ratio, new trapped modes emergence with the vibration mainly focused on the plate with partial electrodes. Besides, owing to the anisotropy, degenerated trapped modes have different resonant frequencies and the frequency discrepancy between them will become smaller for higher modes. Finally, the influence of multiple ring electrodes is investigated, and the qualitative analysis and quantitative results demonstrate that multiple ring electrodes will lead to a more uniform mass sensitivity compared with a single ring electrode. The outcome is widely applicable, which can provide theoretical guidance for the structural design and manufacturing of quartz resonators, as well as a thorough interpretation about the underlying physical mechanism.

Determining material properties of components with complex shapes using Resonant Ultrasound Spectroscopy

In this paper, Resonance Ultrasound Spectroscopy (RUS) is used to invert the elastic properties of components with complex shapes from a limited selection of mode matched frequencies. RUS is based on the unique relationship between the vibrational resonance response (resonance frequencies and mode shapes) of a component and its set of material properties, geometry, and boundary conditions. The study is conducted on 3 nominally identical components that have geometric imperfections due to the manufacturing process. The study reveals that an accurate characterization of the shape is critical to the success of the RUS process. The RUS process presented in this paper also relies on matching measured and computed vibrational mode shapes, in addition to the resonance frequencies. This strategy enables using high-order modes with complex modal information, in a frequency range with high modal density where modal identification based on the resonance frequencies alone would be challenging. This approach contrasts with the traditional approach of matching a number of resonance frequencies in sequential order. With the proposed approach, the elastic constants for the three components are similar, consistent with values found in the literature, and lead to small error of the RUS inversion, despite variations in the shapes and the resulting changes in the resonance response of the components.

Phase Shift Strategies in Phase Shifting Time Averaging Interferometry

Classical time-averaging interferometry technique is widely used for MEMS/MOEMS dynamic behavior investigations. It provides useful information on resonant vibration frequencies and mode shapes in the form of Bessel fringe images. To evaluate the information on objects’ vibration amplitude distribution, one needs to further process Bessel fringe images (besselograms). Among different available approaches to besselograms processing, Temporal Phase Shifting is of particular interest since it provides the most accurate results. However, it requires an additional cumbersome correction routine via specially designed look-up-tables fitted to the calculation algorithm used. In this paper, we numerically investigate the possibility of reducing Bessel images argument (phase) evaluation error by different phase shifting strategies. Two different 5 step algorithms are investigated for that purpose. Additionally, a quick and robust correction procedure based on evaluated phase distribution is presented. It allows us to speed up the calculations significantly. Moreover, such an approach is much less susceptible to noise in comparison to a look-up-table solution.

Multiscale optimisation of resonant frequencies for lattice-based additive manufactured structures

This paper introduces a novel methodology for the optimisation of resonant frequencies in three-dimensional lattice structures. The method uses a multiscale approach in which the homogenised material properties of the lattice unit cell are defined by the spatially varying lattice parameters. Material properties derived from precomputed simulations of the small scale lattice are projected onto response surfaces, thereby describing the large-scale metamaterial properties as polynomial functions of the small-scale parameters. Resonant frequencies and mode shapes are obtained through the eigenvalue analysis of the large-scale finite element model which provides the basis for deriving the frequency sensitivities. Frequency tailoring is achieved by imposing constraints on the resonant frequency for a compliance minimisation optimisation. A sorting method based on the Modal Assurance Criterion allows for specific mode shapes to be optimised whilst simultaneously reducing the impact of localised modes on the optimisation. Three cases of frequency constraints are investigated and compared with an unconstrained optimisation to demonstrate the algorithms applicability. The results show that the optimisation is capable of handling strict frequency constraints and with the use of the modal tracking can even alter the original ordering of the resonant mode shapes. Frequency tailoring allows for improved functionality of compliance-minimised aerospace components by avoiding resonant frequencies and hence dynamic stresses.

Investigation of resonant and energy harvesting characteristics of piezoelectric fiber composite bimorphs

This study investigates the resonant and energy harvesting characteristics of a piezoelectric fiber composite bimorph by using theoretical and experimental techniques, including the full-field optical technique, for the purpose of getting a new perspective on the mechanical and electrical energy transformation ability of piezoelectric composites. A laser Doppler vibrometer, impedance analysis, fotonic sensor, and polyvinylidene fluoride sensor are used in the experiments. Furthermore, an full-field optical technique is also used to measure the resonant frequencies and mode shapes of in-plane and out-of-plane vibrations. Finally, the electrical voltage of the piezoelectric fiber composite bimorph with shaker excitation is measured and used for a light-emitting diode. The shaker input voltage, output voltage of the piezoelectric structure, and light intensity are all measured to find out the power generation capacity of piezoelectric composite. The relationship between the energy harvesting ability and excitation methods, the input voltage value, and frequency are obtained through analyses. From the experiment results, it is indicated that the most efficient modes are affected by the excitation method and the location of an excitation point of the shaker. This study helps in understanding the resonant characteristics as well as the energy harvesting capability of piezoelectric composite.

Exact Mode Shapes of T-shaped and Overhang-shaped Microcantilevers

Resonance frequencies and mode shapes of microcantilevers are of important interest in micro-mechanical systems for enhancing the functionality and applicable range of the cantilevers in vibration transducing, energy harvesting, and highly sensitive measurement. In this study, using the Euler-Bernoulli theory for beam, we figured out the exact mode shapes of cantilevers of varying widths such as the overhang- or T-shaped cantilevers. The obtained mode shapes have been shown to significantly deviate from the approximate forms of a rectangular cantilever that are commonly used in mechanics and physics. They were then used to figure out the resonance frequencies of the cantilever. The analytical solutions have been confirmed by using the finite element method simulations with very low deviation. This study suggested a method for correctly obtaining the resonance frequency of microcantilevers with complicated dimensions, such as the doubly clamped cantilever with the undercut, with the overhangs at the clamped positions, or with an attached mass in the middle.

Resonant ultrasound spectroscopy: Sensitivity analysis for anisotropic materials with hexagonal symmetry

Resonance ultrasound spectroscopy (RUS) is an experimental method for measuring the elastic properties of a material. A sample is excited to vibrate, and its resonant frequencies are measured. From the resonant frequencies and mode shapes, a complete set of elastic constants can be extracted. This result is not always reliable, however. In some cases, the resonant frequencies are insensitive to changes in certain elastic constants or their linear combinations. Previous work has been done to characterize these sensitivity issues in materials with isotropic and cubic symmetry. This work examines the sensitivity of elastic constant measurements by the RUS method for materials with hexagonal symmetry, such as titanium-diboride. We investigate the reliability of RUS data and explore supplemental measurements to obtain an accurate and complete set of elastic constants.

Metamodel-based pattern recognition approach for real-time identification of earthquake-induced damage in historic masonry structures

Damage localization/quantification through vibration-based Structural Health Monitoring (SHM) is commonly performed by inverse calibration of a numerical model. Nevertheless, the numerous simulations required in the associated optimization problem pose a daunting obstacle when applied to real-time SHM. Particularly critical are heritage buildings, whose complex geometries often require computationally intensive modellings. In this light, this paper presents a novel earthquake-induced damage identification approach for historic masonry structures. This relies upon the use of a computationally efficient meta-model suited for real-time system identification. The optimization problem is formulated accounting for discrepancies between numerical and experimental resonant frequencies and mode shapes. Damage localization/quantification is enabled by multivariate analyses of continuously identified model parameters. A real medieval tower is presented as a case study, and several damage scenarios are simulated and used for validation. The reported results pave the way for the development of next-generation long-term vibration-based SHM systems with real-time damage identification capabilities.

Theoretical analysis of vibration characteristics of rectangular thin plate fully immersed in fluid with finite dimension

In this paper, a theoretical method is developed to investigate the effect of a finite flow field on free vibration of a rectangular thin plate fully immersed in an incompressible and inviscid liquid. Rayleigh–Ritz method is applied to obtaining resonant frequencies and mode shapes of the wet plate, and different boundary conditions of a rectangular plate are discussed in this study, including fully clamped, free, and cantilever plates. The Galerkin method is utilized to deal with continuity conditions on the interface between plate and fluid and the beam method is applied to constructing mode shapes of a rectangular plate. Numerical results of the finite element method (FEM) are used to compare with the theoretical analysis on resonant frequencies and mode shapes and it shows the high accuracy of the proposed solution. In order to provide a convenient and accurate method to deal with the vibration characteristics of an immersed plate, the convergence of the Galerkin method is discussed and a modified computation method is presented. Finally, the effect of fluid's boundary on the resonating plate and the relation between the flow field and plate are investigated.

Theoretical analysis and experimental measurement of coupling dynamic characteristics for transversely isotropic rectangular plate based on modified FSDT assumption

The theoretical analysis of thick plate vibration behavior has been investigated in the literature, but most of the studies were focused on flexural dynamic characteristics and lack experimental verification. In this study, the analytical solutions based on the superposition method for both the flexural and extensional vibrations are presented to obtain resonant frequencies and associated mode shapes for a transversely isotropic thick rectangular plate. The displacement equilibrium equations and boundary conditions of the modified first-order shear deformation theory (FSDT) are derived by utilizing Hamilton’s principle and the variation method. To verify the validity of the theoretical model, the finite-element method (FEM) and impact experiment results for thick plates are employed in this work. Excellent agreement of resonant frequencies and associated mode shapes is obtained for FEM calculation and theoretical analysis. To excite vibrations of a thick rectangular plate, a steel ball controlled by an electromagnet is utilized. A steel ball is dropped freely from a height of 231 mm on the top of the plate surface, and a transient wave will be generated after the ball impact on the specimen. The frequency spectrum of the thick rectangular plate is constructed by using the fast Fourier transform of the time-domain transient response. The excited resonant frequencies obtained from experimental measurement are compared with theoretical results. The comparisons show that the modified FSDT provides an excellent prediction of the resonant frequencies and mode shapes for the dynamic characteristics of thick rectangular plates.

Harmonic Standing-Wave Excitations of Simply-Supported Isotropic Solid Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response

The vibration of a solid elastic cylinder is one of the classical applied problems of elastodynamics. Many fundamental forced-vibration problems involving solid elastic cylinders have not yet been studied or solved using the full three-dimensional (3D) theory of linear elasticity. One such problem is the steady-state forced-vibration response of a simply-supported isotropic solid elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surface. In this paper, we exploit certain recently-obtained particular solutions to the Navier–Lamé equation and exact matrix algebra to construct an exact closed-form 3D elastodynamic solution to the problem. The method of solution is direct and demonstrates a general approach that can be applied to solve other similar forced-vibration problems involving elastic cylinders. The obtained analytical solution is then applied to a specific numerical example, where it is used to determine the frequency response of the displacement field in some low wave number excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a subset of the natural frequencies of the simply-supported cylinder. The analytical solution is also used to compute the resonant mode shapes in some selected asymmetric excitation cases. The studied problem is of general interest both as an exactly-solvable 3D elastodynamics problem and as a benchmark forced-vibration problem involving a solid elastic cylinder.